Remark 2.5 In comparison to the localized perturbations studied by [1]}, [2]}, the constant shift of the viscous shock (i.e. Y-shift-p) in this paper is generated essentially by the periodic perturbations and depends additionally on the nonlinearities of the system NS. Thus, the formula Y-shift-p which is expressed in Eulerian coordinates is different from that in Lagrangian coordinates (see [3]}).
It is interesting to find a time-independent formula to compute the limit \( Y_{\infty ,p} \) , which can help understand more clearly the influence of periodic perturbations on the shock stability. However, this is still open except for some 1-d scalar models such as the inviscid convex conservation laws and the viscous Burgers' equation; see [4]}, [5]}, [6]}.

